Selected works. Vol.3 by Ibragimov N.
By Ibragimov N.
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Example text
Let us find the defect δ(Vn , Gr ) of a group of conformal transformations. 5)). 8). 3, the invariant family of spaces Gr (Vn ) represents a one-parameter family of n-dimensional Riemannian spaces in the given case. Let us demonstrate that the family Gr (Vn ) is governed by equations gip (x) yip = y11 g11 (x) (i = 1, . . , n; p = 2, . . 2) where gij (x) is the metric tensor of the space Vn (it is assumed that g11 (x) = 0). One can take y11 as an arbitrary parameter of the family Gr (Vn ). 2) hold when yij = gij (x).
R ) and ν = (ν1 , . . , νr ) are linearly independent. One can easily verify that in this assumptions rank ||ξαi,j + ξαj,i || = 2, so that δ(Vn , Gr ) = 2 according to Eq. 8). 1. Consider a flat n-dimensional space Sn (n > 3) with the metric tensor gij = δij (i, j = 1, . . , n). 1) for every value of α are written in the form ∂ξ k ∂ξ l + = µδkl ∂xl ∂xk (k, l = 1, . . , n), ∂ξ n ∂ξ k + = 0 (k = 1, . . H. IBRAGIMOV SELECTED WORKS, VOL. III Throughout this section the indices k, l run over the values from 1 to n − 1.
9) is a subgroup of a group of conformal transformations in the space Vn . 2) generate a Lie algebra as well. 9). 2) is conformally invariant. 2) with the higher coefficients g ij (x), ij g¯ (x) where gij (x), g¯ij (x) are elements of the corresponding inverse matrices so that gik g jk = δij and g¯ik g¯jk = δij . 2) whatever values the coefficients ai (x) and functions ψ(x, u) take. 2), to which one and the same Riemannian space Vn corresponds, will be further referred to as equations in space Vn .